Optimized formulas of the gravitational field of a vertical cylindrical prism

Modeling the gravitational effects of the topography and other layers of the Earth is the basis not only for gravity corrections and reductions in geodesy but also for gravity inversions and interpretations in geophysics. Previous formulas of the Gravitational Potential (GP), Gravitational Vector (GV), and Gravitational Gradient Tensor (GGT) of a vertical cylindrical prism were derived from complex conversion relations, the result of which is relatively complicated. In this contribution, we are able to optimize such formulas through a particular geometrical relation between the computation point and integration point. The consistency between our newly derived formulas and previous formulas is confirmed analytically. By extending the cylindrical prism to a cylindrical shell, the analytical formulas of the GP, GV, and GGT of a cylindrical shell are derived when the computation point is located on the polar axis. Based on these analytical formulas, a cylindrical shell benchmark is put forward to evaluate the numerical properties of the cylindrical prism, that is to discrete a whole cylindrical shell into cylindrical prisms. For actual numerical calculations, we propose to approximate the cylindrical prism with a second-order 3D Taylor series expansion. Beyond the improved simplicity, our optimized formulas help to save computation time (particularly for the GGT up to 20%). Numerical results reveal that when the computation point’s vertical distance changes, the relative and absolute errors are symmetric with respect to the center vertical distance of the cylindrical shell. Using the second-order 3D Taylor series expansion method provides sufficient computation precision, i.e. all relative errors of the GP, GV, and GGT are smaller than 10⁻² in the numerical experiments. The new expressions for the GP, GV, and GGT of a vertical cylindrical prism using the second-order 3D Taylor series expansion and a cylindrical shell are provided for practical applications of gravity forward modeling in the Python language at the GitHub website https://www.github.com/xiaoledeng/optimized-formulas-of-gp-gv-ggt.